HOW TO FIND THE INVERSE OF A QUADRATIC FUNCTION IN STANDARD FORM

For each quadratic function, complete the square and then determine the equation of the inverse.

Problem 1 :

f(x) = x² + 6x + 15

Solution :

f(x) = x² + 6x + 15

Let f(x) = y

y = x² + 6x + 15

Interchange x and y.

x = y² + 6y + 15

x = y² + 2 ⋅ y ⋅ 3  + 3² - 3²+ 15

x = (y + 3)² + 6

x - 6 = (y + 3)²

Take square root on both sides,

√(x - 6) = (y + 3)

y + 3 = ± √(x - 6)

y = -3 ± √(x - 6)

Replace y by f^-1(x) = -3 ± √x - 6.

Problem 2 :

f(x) = 2x² + 24x - 3

Solution :

f(x) = 2x² + 24x - 3

 Let f(x) = y

y = 2x² + 24x - 3

Interchange x and y.

x = 2y² + 24y - 3

x = 2(y² + 12y) - 3

x = 2(y² + 2 ⋅ y ⋅ 6 + 6² - 6²) - 3

x = 2(y + 6)² - 39

x + 39 = 2(y + 6)²

(x + 39)/2 = (y + 6)²

√(x + 39)/2 = y + 6

y = - 6 ± √(x + 39)/2                

Replace y by f^-1(x) = - 6 ± √(x + 39)/2.          

Problem 3 :

f(x) = x² + 2x - 24

Solution :

f(x) = x² + 2x - 24

Let f(x) = y

y = x² + 2x - 24

Interchange x and y.

x = y² + 2y - 24

x = y² + 2y + (2/2)² - 24

= (y² + 2 ⋅ y ⋅1 + 1² - 1²) - 24

x = (y + 1)² - 25

x + 25 = (y + 1)²

Take square root on both sides,

√x + 25 = (y + 1)

y + 1 = ± √x + 25

y = -1 ± √x + 25

Replace y by f^-1(x) = -1 ± √x + 25.

Problem 4 :

f(x) = p² + 12p - 54

Solution :

f(x) = p² + 12p - 54

Let f(x) = y

y = p² + 12p - 54

Interchange x and y.

x = p² + 12p - 54

x = p² + 2 ⋅ p ⋅ 6 + 6² - 6² - 54

x = (y + 6)²- 54 - 36

x = (y + 6)² - 90

x + 90 = (y + 6)²

Take square root on both sides,

√x + 90 = (y + 6)

y + 6 = ± √(x + 90)

y = -6 ± √x + 90

Replace y by f^-1(x) = -6 ± √x + 90.

Problem 5 :

f(x) = x² - 8x + 15

Solution :

f(x) = x² - 8x + 15

Let f(x) = y

y = x² - 8x + 15

Interchange x and y.

x = y² - 8y + 15

x = y² - 2 ⋅ y ⋅ 4 + 4² - 4² + 15

x = (y - 4)² - 16 + 15

x + 1 = (y - 4)²

Take square root on both sides,

√x + 1 = (y - 4)

y - 4 = ± √x + 1

y = 4 ± √x + 1

Replace y by f^-1(x) = 4 ± √x + 1.

Problem 6 :

f(x) = r² + 18r + 56

Solution :

f(x) = x² + 18x + 56

Let f(x) = y

y = x² + 18x + 56

Interchange x and y.

x = y² + 18y + 56

x = y² + 2 ⋅ y ⋅ 9 + 9² - 9² + 56

x = (y + 9)² + 56 - 81

x = (y + 9)² - 25

x + 25 = (y + 9)²

Take square root on both sides,

√(x + 25) = (y + 9)

y + 9 = ± √(x + 25)

y = -9 ± √(x + 25)

Replace y by f^-1(x) = -9 ± √x + 25.